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Theorem latleeqm1 14513
Description: Less-than-or-equal-to in terms of meet. (df-ss 3336 analog.) (Contributed by NM, 7-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latleeqm1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )

Proof of Theorem latleeqm1
StepHypRef Expression
1 latmle.b . . . . . . 7  |-  B  =  ( Base `  K
)
2 latmle.l . . . . . . 7  |-  .<_  =  ( le `  K )
31, 2latref 14487 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  X  .<_  X )
433adant3 978 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  .<_  X )
54biantrurd 496 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  .<_  X  /\  X  .<_  Y ) ) )
6 simp1 958 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Lat )
7 simp2 959 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
8 simp3 960 . . . . 5  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
9 latmle.m . . . . . 6  |-  ./\  =  ( meet `  K )
101, 2, 9latlem12 14512 . . . . 5  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  X  e.  B  /\  Y  e.  B
) )  ->  (
( X  .<_  X  /\  X  .<_  Y )  <->  X  .<_  ( X  ./\  Y )
) )
116, 7, 7, 8, 10syl13anc 1187 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .<_  X  /\  X  .<_  Y )  <-> 
X  .<_  ( X  ./\  Y ) ) )
125, 11bitrd 246 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  X  .<_  ( X  ./\  Y )
) )
131, 2, 9latmle1 14510 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  .<_  X )
1413biantrurd 496 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  ( X 
./\  Y )  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
1512, 14bitrd 246 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) ) ) )
16 latpos 14483 . . . 4  |-  ( K  e.  Lat  ->  K  e.  Poset )
17163ad2ant1 979 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  K  e.  Poset )
181, 9latmcl 14485 . . 3  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  ./\  Y
)  e.  B )
191, 2posasymb 14414 . . 3  |-  ( ( K  e.  Poset  /\  ( X  ./\  Y )  e.  B  /\  X  e.  B )  ->  (
( ( X  ./\  Y )  .<_  X  /\  X  .<_  ( X  ./\  Y ) )  <->  ( X  ./\ 
Y )  =  X ) )
2017, 18, 7, 19syl3anc 1185 . 2  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( ( X 
./\  Y )  .<_  X  /\  X  .<_  ( X 
./\  Y ) )  <-> 
( X  ./\  Y
)  =  X ) )
2115, 20bitrd 246 1  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<_  Y  <->  ( X  ./\ 
Y )  =  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   lecple 13541   Posetcpo 14402   meetcmee 14407   Latclat 14479
This theorem is referenced by:  latleeqm2  14514  latnlemlt  14518  latabs2  14522  atnle  30189  2llnmat  30395  llnmlplnN  30410  dalem25  30569  2lnat  30655  lhpm0atN  30900  lhpmatb  30902  cdleme1  31098  cdleme5  31111  cdleme20d  31183  cdleme22e  31215  cdleme22eALTN  31216  cdleme23b  31221  cdleme32e  31316  doca2N  31998  djajN  32009  dihglblem5aN  32164  dihmeetbclemN  32176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-glb 14437  df-meet 14439  df-lat 14480
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